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反问题、不适定性和正则化方法

(2012-07-04 15:16:43)
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反问题(inverse problems)、不适定性(ill-posedness,Hadamard)和正则化方法(regularization method)

 

给定多项式的赋值为正问题,插值为反问题(inverse problems)。若问题的解/答案不连续地依

赖于原始数据,称其为反问题,在Hadamard意义下是不适定的!

反问题的一个共同特征就是所谓的“不适定性(ill-posedness)”,适定和不适定的概念是Hadamard为描述数学物理问题与定解条件的合理搭配,于20世纪初引入的!

把求数学物理反问题的稳定近似解的方法称为正则化方法或策略(regularization method/strategy,Tikhonov和Phillips),有Tikhonov和Phillips在20世纪60年代出分别独立提出!

 

 

Andrey Nikolayevich Tikhonov

Andrey Tikhonov
Born (1906-10-30)30 October 1906
Gzhatsk, Russian Empire
Died November 8, 1993(1993-11-08) (aged 87)
Moscow, Russia
Citizenship  Russian Empire
 Soviet Union
 Russia
Nationality Russia
Fields mathematics
Institutions Moscow State University
Alma mater Moscow State University
Doctoral advisor Pavel Alexandrov
Doctoral students Alexander Samarsky
Known for important contributions to topology, functional analysis, mathematical physics, ill-posed problems; Tychonoff spaces, Tychonoff's theorem, Tikhonov regularization, Tikhonov's theorem (dynamical systems).

Andrey Nikolayevich Tikhonov (Russian: Андре́й Никола́евич Ти́хонов; October 30, 1906, Gzhatsk – November 8, 1993, Moscow) was a Soviet and Russian mathematician known for important contributions to topology, functional analysis, mathematical physics, and ill-posed problems. He was also one of the inventors of the magnetotellurics method in geology. Tikhonov's surname is also transliterated as "Tychonoff", because he originally published in German.

Biography

Born near Smolensk, he studied at the Moscow State University where he received Ph.D. in 1927 under direction of Pavel Sergeevich Alexandrov. In 1933 he was appointed as a professor at Moscow State University. He became a corresponding member of the USSR Academy of Sciences on 29 January 1939 and a full member of the USSR Academy of Sciences on 1 July 1966.

Research work

Tikhonov worked in a number of different fields in mathematics. He made important contributions to topology, functional analysis, mathematical physics, and certain classes of ill-posed problems. Tikhonov regularization, one of the most widely used methods to solve ill-posed inverse problems, is named in his honor. He is best known for his work on topology, including the metrization theorem he proved in 1926, and the Tychonoff's theorem, which states that every product of arbitrarily many compact topological spaces is again compact. In his honor, completely regular topological spaces are also named Tychonoff spaces.

In mathematical physics, he proved the fundamental uniqueness theorems for the heat equation[1] and studied Volterra integral equations.

In asymptotical analysis, he founded the theory of asymptotic analysis for differential equations with small parameter in the leading derivative.[2]

Organizer work

Tikhonov played the leading role in founding the Faculty of Computational Mathematics and Cybernetics of Moscow State University and served as its first dean during the period of 1970–1990.

Memorial board of A.N. Tikhonov on the MSU Second Humanities Building where the Faculty of Computational Mathematics and Cybernetics is located

Awards

Tikhonov received numerous honors and awards for his work, including the Lenin Prize (1966) and the Hero of Socialist Labor (1954, 1986).

Publications

Books

  • A.G. Sveshnikov, A.N. Tikhonov, The Theory of Functions of a Complex Variable, Mir Publishers, English translation, 1978.
  • A.N. Tikhonov, V.Y. Arsenin, Solutions of Ill-Posed Problems, Winston, New York, 1977. ISBN 0-470-99124-0.
  • A.N. Tikhonov, A.V. Goncharsky, Ill-posed Problems in the Natural Sciences, Oxford University Press, Oxford, 1987. ISBN 0-8285-3739-9.
  • A.N. Tikhonov, A.A. Samarskii, Equations of Mathematical Physics, Dover Publications, 1990. ISBN 0-486-66422-8.
  • A.N. Tikhonov, A.V. Goncharsky, V.V. Stepanov, A.G. Yagola, Numerical Methods for the Solution of Ill-Posed Problems, Kluwer, Dordrecht, 1995. ISBN 0-7923-3583-X.
  • A.N. Tikhonov, A.S. Leonov, A.G. Yagola. Nonlinear Ill-Posed Problems, Chapman and Hall, London, Weinheim, New York, Tokyo, Melbourne, Madras, V. 1-2, 1998. ISBN 0-412-78660-5.

Papers

  1. ^ A. Tychonoff (1935). "Théorèmes d'unicité pour l'équation de la chaleur". Mathematical Sbornik 42 (2): 199–216. http://mi.mathnet.ru/eng/msb6410. 
  2. ^ A. N. Tikhonov (1952). "Systems of Differential Equations Containing Small Parameters in the Derivatives". Mathematical Sbornik 31(73):3: 575–586. http://mi.mathnet.ru/eng/msb5548. 

External links

Persondata
Name Tychonoff, Andrey Nikolayevich
Alternative names
Short description
Date of birth 30 October 1906
Place of birth Gzhatsk, Russian Empire
Date of death 8 November 1993
Place of death Moscow, Russia

 

 

吉洪诺夫正则化(Tikhonov regularization)http://blog.sina.com.cn/s/blog_4b700c4c0102e4qq.html

 

 

 

Tikhonov regularization

Regression analysis

Linear regression.svg

Models

Linear regression
Simple regression
Ordinary least squares
Polynomial regression
General linear model


Generalized linear model
Discrete choice · Logistic regression
Multinomial logit · Mixed logit
Probit · Multinomial probit
Ordered logit · Ordered probit
Poisson


Multilevel model
Fixed effects · Random effects
Mixed model


Nonlinear regression
Nonparametric · Semiparametric
Robust · Quantile · Isotonic
Principal components · Least angle
Local · Segmented


Errors-in-variables

Estimation

Least squares · Ordinary least squares
Linear (math) · Partial · Total
Generalized · Weighted
Non-linear · Iteratively reweighted
Ridge regression · LASSO


Least absolute deviations
Bayesian · Bayesian multivariate

Background

Regression model validation
Mean and predicted response
Errors and residuals
Goodness of fit · Studentized residual
Gauss–Markov theorem

Tikhonov regularization, named for Andrey Tikhonov, is the most commonly used method of regularization of ill-posed problems. In statistics, the method is known as ridge regression, and, with multiple independent discoveries, it is also variously known as the Tikhonov–Miller method, the Phillips–Twomey method, the constrained linear inversion method, and the method of linear regularization. It is related to the Levenberg–Marquardt algorithm for non-linear least-squares problems.

When the following problem is not well posed (either because of non-existence or non-uniqueness of x )

A\mathbf{x}=\mathbf{b},

then the standard approach is known as linear least squares and seeks to minimize the residual

\|A\mathbf{x}-\mathbf{b}\|^2

where \left \| \cdot \right \| is the Euclidean norm. This may be due to the system being overdetermined or underdetermined (A may be ill-conditioned or singular). In the latter case this is no better than the original problem. In order to give preference to a particular solution with desirable properties, the regularization term is included in this minimization:

\|A\mathbf{x}-\mathbf{b}\|^2+ \|\Gamma \mathbf{x}\|^2

for some suitably chosen Tikhonov matrix, \Gamma . In many cases, this matrix is chosen as the identity matrix \Gamma= I , giving preference to solutions with smaller norms. In other cases, highpass operators (e.g., a difference operator or a weighted Fourier operator) may be used to enforce smoothness if the underlying vector is believed to be mostly continuous. This regularization improves the conditioning of the problem, thus enabling a numerical solution. An explicit solution, denoted by \hat{x} , is given by:

\hat{x} = (A^{T}A+ \Gamma^{T} \Gamma )^{-1}A^{T}\mathbf{b}

The effect of regularization may be varied via the scale of matrix \Gamma . For \Gamma = 0 this reduces to the unregularized least squares solution provided that (ATA)−1 exists.

Bayesian interpretation

Although at first the choice of the solution to this regularized problem may look artificial, and indeed the matrix \Gamma seems rather arbitrary, the process can be justified from a Bayesian point of view. Note that for an ill-posed problem one must necessarily introduce some additional assumptions in order to get a stable solution. Statistically, the prior probability distribution of x is sometimes taken to be a multivariate normal distribution. For simplicity here, the following assumptions are made: the means are zero; their components are independent; the components have the same standard deviation \sigma _x . The data are also subject to errors, and the errors in b are also assumed to be independent with zero mean and standard deviation \sigma _b . Under these assumptions the Tikhonov-regularized solution is the most probable solution given the data and the a priori distribution of x , according to Bayes' theorem.[1] The Tikhonov matrix is then \Gamma = \alpha I for Tikhonov factor \alpha = \sigma _b / \sigma _x .

If the assumption of normality is replaced by assumptions of homoskedasticity and uncorrelatedness of errors, and if one still assumes zero mean, then the Gauss–Markov theorem entails that the solution is the minimal unbiased estimate.[citation needed]

Generalized Tikhonov regularization

For general multivariate normal distributions for x and the data error, one can apply a transformation of the variables to reduce to the case above. Equivalently, one can seek an x to minimize

\|Ax-b\|_P^2 + \|x-x_0\|_Q^2\,

where we have used \left \| x \right \|_Q^2 to stand for the weighted norm x^T Q x (compare with the Mahalanobis distance). In the Bayesian interpretation P is the inverse covariance matrix of b , x_0 is the expected value of x , and Q is the inverse covariance matrix of x . The Tikhonov matrix is then given as a factorization of the matrix  Q = \Gamma^T \Gamma (e.g. the Cholesky factorization), and is considered a whitening filter.

This generalized problem can be solved explicitly using the formula

x_0 + (A^T PA + Q)^{-1} A^T P(b-Ax_0).\,

Regularization in Hilbert space

Typically discrete linear ill-conditioned problems result from discretization of integral equations, and one can formulate a Tikhonov regularization in the original infinite dimensional context. In the above we can interpret A as a compact operator on Hilbert spaces, and x and b as elements in the domain and range of A . The operator A^* A + \Gamma^T \Gamma is then a self-adjoint bounded invertible operator.

Relation to singular value decomposition and Wiener filter

With \Gamma = \alpha I , this least squares solution can be analyzed in a special way via the singular value decomposition. Given the singular value decomposition of A

A = U \Sigma V^T\,

with singular values \sigma _i , the Tikhonov regularized solution can be expressed as

\hat{x} = V D U^T b

where D has diagonal values

D_{ii} = \frac{\sigma _i}{\sigma _i ^2 + \alpha ^2}

and is zero elsewhere. This demonstrates the effect of the Tikhonov parameter on the condition number of the regularized problem. For the generalized case a similar representation can be derived using a generalized singular value decomposition.

Finally, it is related to the Wiener filter:

\hat{x} = \sum _{i=1} ^q f_i \frac{u_i ^T b}{\sigma _i} v_i

where the Wiener weights are f_i = \frac{\sigma _i ^2}{\sigma _i ^2 + \alpha ^2} and q is the rank of A .

Determination of the Tikhonov factor

The optimal regularization parameter \alpha is usually unknown and often in practical problems is determined by an ad hoc method. A possible approach relies on the Bayesian interpretation described above. Other approaches include the discrepancy principle, cross-validation, L-curve method, restricted maximum likelihood and unbiased predictive risk estimator. Grace Wahba proved that the optimal parameter, in the sense of leave-one-out cross-validation minimizes:

G = \frac{\operatorname{RSS}}{\tau ^2} = \frac{\left \| X \hat{\beta} - y \right \| ^2}{\left[ \operatorname{Tr} \left(I - X (X^T X + \alpha ^2 I) ^{-1} X ^T \right) \right]^2}

where \operatorname{RSS} is the residual sum of squares and \tau is the effective number of degrees of freedom.

Using the previous SVD decomposition, we can simplify the above expression:

\operatorname{RSS} = \left \| y - \sum _{i=1} ^q (u_i ' b) u_i \right \| ^2 + \left \| \sum _{i=1} ^q \frac{\alpha ^ 2}{\sigma _i ^ 2 + \alpha ^ 2} (u_i ' b) u_i \right \| ^2
\operatorname{RSS} = \operatorname{RSS} _0 + \left \| \sum _{i=1} ^q \frac{\alpha ^ 2}{\sigma _i ^ 2 + \alpha ^ 2} (u_i ' b) u_i \right \| ^2

and

\tau = m - \sum _{i=1} ^q \frac{\sigma _i ^2}{\sigma _i ^2 + \alpha ^2}
= m - q + \sum _{i=1} ^q \frac{\alpha ^2}{\sigma _i ^2 + \alpha ^2}

Relation to probabilistic formulation

The probabilistic formulation of an inverse problem introduces (when all uncertainties are Gaussian) a covariance matrix  C_M representing the a priori uncertainties on the model parameters, and a covariance matrix  C_D representing the uncertainties on the observed parameters (see, for instance, Tarantola, 2004 [1]). In the special case when these two matrices are diagonal and isotropic,  C_M = \sigma_M^2 I and  C_D = \sigma_D^2 I , and, in this case, the equations of inverse theory reduce to the equations above, with  \alpha = {\sigma_D}/{\sigma_M} .

History

Tikhonov regularization has been invented independently in many different contexts. It became widely known from its application to integral equations from the work of Tychonoff and David L. Phillips. Some authors use the term Tikhonov–Phillips regularization. The finite dimensional case was expounded by Arthur E. Hoerl, who took a statistical approach, and by Manus Foster, who interpreted this method as a WienerKolmogorov filter. Following Hoerl, it is known in the statistical literature as ridge regression.

See also

References

  • Tychonoff, Andrey Nikolayevich (1943). "Об устойчивости обратных задач [On the stability of inverse problems]". Doklady Akademii Nauk SSSR 39 (5): 195–198. 
  • Tychonoff, A. N. (1963). "О решении некорректно поставленных задач и методе регуляризации [Solution of incorrectly formulated problems and the regularization method]". Doklady Akademii Nauk SSSR 151: 501–504. . Translated in Soviet Mathematics 4: 1035–1038. 
  • Tychonoff, A. N.; V. Y. Arsenin (1977). Solution of Ill-posed Problems. Washington: Winston & Sons. ISBN 0-470-99124-0. 
  • Tikhonov A.N., Goncharsky A.V., Stepanov V.V., Yagola A.G., 1995, Numerical Methods for the Solution of Ill-Posed Problems, Kluwer Academic Publishers.
  • Tikhonov A.N., Leonov A.S., Yagola A.G., 1998, Nonlinear Ill-Posed Problems, V. 1, V. 2, Chapman and Hall.
  • Hansen, P.C., 1998, Rank-deficient and Discrete ill-posed problems, SIAM
  • Hoerl AE, 1962, Application of ridge analysis to regression problems, Chemical Engineering Progress, 58, 54–59.
  • Hoerl, A.E.; R.W. Kennard (1970). "Ridge regression: Biased estimation for nonorthogonal problems". Technometrics 42 (1): 80–86. JSTOR 1271436. 
  • Foster M, 1961, An application of the Wiener-Kolmogorov smoothing theory to matrix inversion, J. SIAM, 9, 387–392
  • Phillips DL, 1962, A technique for the numerical solution of certain integral equations of the first kind, J Assoc Comput Mach, 9, 84–97
  • Press, WH; Teukolsky, SA; Vetterling, WT; Flannery, BP (2007). "Section 19.5. Linear Regularization Methods". Numerical Recipes: The Art of Scientific Computing (3rd ed.). New York: Cambridge University Press. ISBN 978-0-521-88068-8. http://apps.nrbook.com/empanel/index.html#pg=1006. 
  • Tarantola A, 2004, Inverse Problem Theory (free PDF version), Society for Industrial and Applied Mathematics, ISBN 0-89871-572-5
  • Wahba, G, 1990, Spline Models for Observational Data, Society for Industrial and Applied Mathematics
  1. ^ Vogel, Curtis R. (2002). Computational methods for inverse problems. Philadelphia: Society for Industrial and Applied Mathematics. ISBN 0-89871-550-4. 

 

 

 

 

 

 

 

 

 

 

雅克·阿达马

雅克·所罗门·阿达马

雅克·所罗门·阿达马Jacques Solomon Hadamard,1865年12月8日-1963年10月17日)是法国数学家。他最有名的是他的素数定理证明。

他在巴黎高师(École Normale Supérieure)学习。德雷福斯事件他也牵涉在内,此后他活跃于政治,坚定支持为犹太人的事业。

他为偏微分方程创造了适定问题概念。他也给了其名字予论体积的阿达马不等式,还有阿达马矩阵,是阿达马变换所以建基的。量子计算阿达马门使用这个矩阵。

曾受清华大学邀请至中国讲学(同时期者尚有美国数学家维纳),影响了中国近代数学发展。

在阿达马所著的《数学领域的发明心理学》(Psychology of Invention in the Mathematical Field),他用内省来描述数学思维过程。与把认知语言等同的作者截然相反,他描述他的数学思考大部分是无字的,往往有心象伴随著,浓缩了证明的整体思路。

他学生之一是安德烈·韦伊

著作

  • Hadamard, Jacques, "Psychology of Invention in the Mathematical Field". Dover Pubns; November 1990. ISBN 0-486-20107-4

初等几何学教程

深入阅读

  • Life and Work of Jacques Hadamard, Vladimir Maz'ya & T. O. Shaposhnikova, American Mathematical Society, February 1998, hardcover, 574 pages, ISBN 0-8218-0841-9


 

阿达玛

阿达玛(Hadamard,Jacques---Salomon 1865.12.8-1963.10.17)法国数学家,最先证明了素数定理,即当n趋于无穷大时,π(n)是不大于n的素烽的个数。他先后任安西学院(1897-1935年)、巴黎综合工科学校(1912-1935年)和中央工艺制造学院(1920-1935年)的教授。他早期研究复变函数论,对整函数的一般理论以及用级数表示的函数的奇点理论有重要贡献。1896年,他与比利时数学家C·J·普森各自独立地证明了素数定理。他在数学物理偏微分方程方面也取得了重要成果。他的著作《变分法教程》对于泛函分析近代理论的奠定打下了基础,“泛函”一词就是他首先使用的。

 

阿达玛 - 相关理论

泛函分析是研究拓扑线性空间到拓扑线性空间之间满足各种拓扑和代数条件的映射的分支学科。它是20世纪30年代形成的。从变分法、微分方程、积分方程、函数论以及量子物理等的研究中发展起来的,它运用几何学、代数学的观点和方法研究分析学的课题,可看作无限维的分析学。

  姓名:阿达玛 Ha
  

 

damard,Jacques---Salomon
  国家或者地区:法国
  学科:数学家
  发明创造:最先证明素数定理
  简 历
  阿达玛(Hadamard,Jacques---Salomon 1865.12.8-1963.10.17)法国数学家,最先证明了素数定理,即当n趋于无穷大时,π(n)是不大于n的素烽的个数。他先后任安西学院(1897-1935年)、巴黎综合工科学校(1912-1935年)和中央工艺和制造学院(1920-1935年)的教授。他早期研究复变函数论,对整函数的一般理论以及用级数表示的函数的奇点理论有重要贡献。1896年,他与比利时数学家C·J·普森各自独立地证明了素数定理。他在数学物理偏微分方程方面也取得了重要成果。他的著作《变分法教程》对于泛函分析近代理论的奠定打下了基础,“泛函”一词就是他首先使用的。
  图像信号处理中的离散沃尔什-哈达玛变换,就是此人。
  Hadamard变换作为变换编码的一种在视频编码当中使用有很久的历史。在近来的视频编码标准中,Hadamard变换多被用来计算SATD(一种视频残差信号大小的衡量)。
  泛函分析是研究拓扑线性空间到拓扑线性空间之间满足各种拓扑和代数条件的映射的分支学科。它是20世纪30年代形成的。从变分法、微分方程、积分方程、函数论以及量子物理等的研究中发展起来的,它运用几何学、代数学的观点和方法研究分析学的课题,可看作无限维的分析学。
  Hadamard乘积
  两个矩阵作Hadamard乘积,则意味着两个矩阵中对应的元素相乘。与矩阵的乘法不同。
 
 
 

Jacques Hadamard

 
Jacques Hadamard

Jacques Salomon Hadamard
Born (1865-12-08)8 December 1865
Versailles, France
Died 17 October 1963(1963-10-17) (aged 97)
Paris, France
Residence France
Nationality French
Fields Mathematician
Institutions University of Bordeaux
Sorbonne
Collège de France
École Polytechnique
École Centrale
Alma mater École Normale Supérieure
Doctoral advisor C. Émile Picard[1]
Jules Tannery
Doctoral students Maurice René Fréchet
Paul Lévy
Szolem Mandelbrojt
André Weil
Xinmou Wu
Known for Hadamard product
Proof of prime number theorem
Hadamard matrices
Notable awards Grand Prix des Sciences Mathématiques (1892)
Prix Poncelet (1898)
CNRS Gold medal (1956)

Jacques Salomon Hadamard FRS[2] (8 December 1865 – 17 October 1963) was a French mathematician who made major contributions in number theory, complex function theory, differential geometry and partial differential equations.[3][4]


Biography

The son of a teacher, Amédée Hadamard, of Jewish descent, and Claire Marie Jeanne Picard, Hadamard was born in Versailles, France and attended the Lycée Charlemagne and Lycée Louis-le-Grand, where his father taught. In 1884 Hadamard entered the École Normale Supérieure, having been placed first in the entrance examinations both there and at the École Polytechnique. His teachers included Tannery, Hermite, Darboux, Appell, Goursat and Picard. He obtained his doctorate in 1892 and in the same year was awarded the Grand Prix des Sciences Mathématiques for his essay on the Riemann zeta function.

In 1892 Hadamard married Louise-Anna Trénel, also of Jewish descent, with whom he had three sons and two daughters. The following year he took up a lectureship in the University of Bordeaux, where he proved his celebrated inequality on determinants, which led to the discovery of Hadamard matrices when equality holds. In 1896 he made two important contributions: he proved the prime number theorem, using complex function theory (also proved independently by Charles Jean de la Vallée-Poussin); and he was awarded the Bordin Prize of the French Academy of Sciences for his work on geodesics in the differential geometry of surfaces and dynamical systems. In the same year he was appointed Professor of Astronomy and Rational Mechanics in Bordeaux. His foundational work on geometry and symbolic dynamics continued in 1898 with the study of geodesics on surfaces of negative curvature. For his cumulative work, he was awarded the Prix Poncelet in 1898.

After the Dreyfus affair, which involved him personally because his wife was related to Dreyfus, Hadamard became politically active and a staunch supporter of Jewish causes[5] though he professed to be an atheist in his religion.[6]

In 1897 he moved back to Paris, holding positions in the Sorbonne and the Collège de France, where he was appointed Professor of Mechanics in 1909. In addition to this post, he was appointed to chairs of analysis at the École Polytechnique in 1912 and at the École Centrale in 1920, succeeding Jordan and Appell. In Paris Hadamard concentrated his interests on the problems of mathematical physics, in particular partial differential equations, the calculus of variations and the foundations of functional analysis. He introduced the idea of well-posed problem and the method of descent in the theory of partial differential equations, culminating in his seminal book on the subject, based on lectures given at Yale University in 1922. Later in his life he wrote on probability theory and mathematical education.

Hadamard was elected to the French Academy of Sciences in 1916, in succession to Poincaré, whose complete works he helped edit. He was elected a foreign member of the Academy of Sciences of the USSR in 1929. He visited the Soviet Union in 1930 and 1934 and China in 1936 at the invitation of Soviet and Chinese mathematicians.

Hadamard stayed in France at the beginning of the Second World War and escaped to southern France in 1940. The Vichy government permitted him to leave for the United States in 1941 and he obtained a visiting position at Columbia University in New York. He moved to London in 1944 and returned to France when the war ended in 1945.

He was awarded the CNRS Gold medal for his lifetime achievements in 1956. He died in Paris in 1963, aged ninety-seven.

Hadamard's students included Maurice Fréchet, Paul Lévy, Szolem Mandelbrojt and André Weil.

On creativity

In his book Psychology of Invention in the Mathematical Field, Hadamard uses introspection to describe mathematical thought processes. In sharp contrast to authors who identify language and cognition, he describes his own mathematical thinking as largely wordless, often accompanied by mental images that represent the entire solution to a problem. He surveyed 100 of the leading physicists of the day (approximately 1900), asking them how they did their work.

Hadamard described the experiences of the mathematicians/theoretical physicists Carl Friedrich Gauss, Hermann von Helmholtz, Henri Poincaré and others as viewing entire solutions with “sudden spontaneousness”.[7]

Hadamard described the process as having four steps of the five-step Graham Wallas creative process model, with the first three also having been put forth by Helmholtz:[8] Preparation, Incubation, Illumination, and Verification.

See also

References

  1. ^ Hadamard, J. (1942). "Emile Picard. 1856-1941". Obituary Notices of Fellows of the Royal Society 4 (11): 129–126. DOI:10.1098/rsbm.1942.0012.  edit
  2. ^ Cartwright, M. L. (1965). "Jacques Hadamard. 1865-1963". Biographical Memoirs of Fellows of the Royal Society 11: 75–26. DOI:10.1098/rsbm.1965.0005.  edit
  3. ^ O'Connor, John J.; Robertson, Edmund F., "Jacques Hadamard", MacTutor History of Mathematics archive, University of St Andrews, http://www-history.mcs.st-andrews.ac.uk/Biographies/Hadamard.html .(or, see: this Webcite "backup" copy, archived from the original)
  4. ^ Jacques Hadamard at the Mathematics Genealogy Project
  5. ^ Hadamard, Jacques (1954). An essay on the psychology of invention in the mathematical field / by Jacques Hadamard. New York: Dover Publications. ISBN 0-486-20107-4. 
  6. ^ Hadamard on Hermite
  7. ^ Hadamard, 1954, pp. 13-16.
  8. ^ Hadamard, 1954, p. 56.

Publications

Further reading

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